Average Error: 29.6 → 0.4
Time: 32.2s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -306416.6377200472052209079265594482421875:\\ \;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 4.236193905385104490052934015503760534216 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -306416.6377200472052209079265594482421875:\\
\;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\

\mathbf{elif}\;-2 \cdot x \le 4.236193905385104490052934015503760534216 \cdot 10^{-7}:\\
\;\;\;\;x \cdot \left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2331777 = 2.0;
        double r2331778 = 1.0;
        double r2331779 = -2.0;
        double r2331780 = x;
        double r2331781 = r2331779 * r2331780;
        double r2331782 = exp(r2331781);
        double r2331783 = r2331778 + r2331782;
        double r2331784 = r2331777 / r2331783;
        double r2331785 = r2331784 - r2331778;
        return r2331785;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2331786 = -2.0;
        double r2331787 = x;
        double r2331788 = r2331786 * r2331787;
        double r2331789 = -306416.6377200472;
        bool r2331790 = r2331788 <= r2331789;
        double r2331791 = 2.0;
        double r2331792 = exp(r2331788);
        double r2331793 = 1.0;
        double r2331794 = r2331792 + r2331793;
        double r2331795 = sqrt(r2331794);
        double r2331796 = sqrt(r2331795);
        double r2331797 = r2331791 / r2331796;
        double r2331798 = r2331797 / r2331796;
        double r2331799 = r2331798 / r2331795;
        double r2331800 = r2331799 - r2331793;
        double r2331801 = 4.2361939053851045e-07;
        bool r2331802 = r2331788 <= r2331801;
        double r2331803 = 0.33333333333333337;
        double r2331804 = r2331787 * r2331787;
        double r2331805 = r2331803 * r2331804;
        double r2331806 = r2331793 - r2331805;
        double r2331807 = r2331787 * r2331806;
        double r2331808 = 5.551115123125783e-17;
        double r2331809 = r2331804 * r2331808;
        double r2331810 = r2331804 * r2331809;
        double r2331811 = r2331807 - r2331810;
        double r2331812 = r2331802 ? r2331811 : r2331800;
        double r2331813 = r2331790 ? r2331800 : r2331812;
        return r2331813;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* -2.0 x) < -306416.6377200472 or 4.2361939053851045e-07 < (* -2.0 x)

    1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{2}{\color{blue}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}}} - 1\]

    4. Applied associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}}} - 1\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{\frac{2}{\sqrt{\color{blue}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\]

    7. Applied sqrt-prod0.1

      \[\leadsto \frac{\frac{2}{\color{blue}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}} \cdot \sqrt{\sqrt{1 + e^{-2 \cdot x}}}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\]

    8. Applied associate-/r*0.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\]

    if -306416.6377200472 < (* -2.0 x) < 4.2361939053851045e-07

    1. Initial program 58.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -306416.6377200472052209079265594482421875:\\ \;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 4.236193905385104490052934015503760534216 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{\sqrt{e^{-2 \cdot x} + 1}}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))